Smoothness of functions and Fourier coefficients

Abstract

We consider functions represented as trigonometric series with general monotone Fourier coefficients. The main result of the paper is the equivalence of the $L_p$ modulus of smoothness, $1< p<\infty$, of such functions to certain sums of their Fourier coefficients. As applications, for such functions we give a description of the norm in the Besov space and sharp direct and inverse theorems in approximation theory.